If I understand correctly, noise cancelling headphones isolate a certain frequency or perhaps a sample of a frequency along an given timeframe. If this is true, then the isolated frequency can be amplified as well as dupilcated. I also assume that these devices have to have some sort of microphones for
audio input. Headphones take the frequencies that they would normally cancel and amplify them so you could hear your friend ask you a question at twice the normal volume so you would end up answering them in a more reduced volume. You could also tune into other conversations when others think you are oblivious. A button would be included to revert their operation to noise cancelling cancelling cancelling headphones with twice the cancelling power.

I did a long explanation involving FFTs, but now I'm thinking the idea is nothing more than an option to turn noise cancelling off or to amplify background noise by a factor of two - so a fancy mute button. I once had an amp. that had a 10db mute on the remote - far more elegant than on/off.

I don't think so. I think the idea considers noise and wanted sounds to occupy non-overlapping frequency bands, and existing noise-canceling headphones to be able to magically identify which frequencies contain which type of sound. My understanding is that they usually work by trying to cancel all external sounds, but I could be wrong about that.

I'm not sure what you mean about the mute button. Function generators commonly have buttons (or knobs) to reduce the output amplitude by (up to) 10, 20, or 30 dB, but this is considered "attenuation" rather than "muting". Doesn't the mute button on a TV or radio just disable the audio output entirely?